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Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

Published online by Cambridge University Press:  20 November 2018

M. Edelstein
M. Edelstein*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 933 - 938
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Asplund, E., Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47.Google Scholar
2. Bessaga, C. and Pekzynski, A., On extreme points in separable conjugate spaces, Israel J. Math. 4 (1966), 262–264.Google Scholar
3. Edelstein, M., Intersections by hyperplanes﹜ Israel J. Math. 7 (1969), 90–94.Google Scholar
4. Edelstein, M., Hyper planes and lines associated with families of compact sets in locally convex spaces, Math. Scand. 25 (1969), 25–30.Google Scholar
5. Edelstein, M. and Kelly, L. M., Bisecants of finite collections of sets in linear spaces, Can. J. Math. 18 (1966), 375–380.Google Scholar
6. Edelstein, M., Herzog, F., and Kelly, L. M., A further theorem of the Sylvester type, Proc. Amer. Math. Soc. 14 (1963), 359–363.Google Scholar
7. Kelly, L. M., Linear transversals, Proc. London Math. Soc. 16 (1966), 264–274.Google Scholar
8. Kothe, G., Topologische linear e Raume. I. Die Grundlehren der mathematischen Wissenschaften, Band 107 (Springer-Verlag, Berlin-Gottingen-Heidelberg, 1960).Google Scholar
9. Namioka, I., Neighborhoods of extreme points, Israel J. Math. 5 (1967), 145–152.Google Scholar